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There are three methods of screw extraction when the screw heads are hopelessly damaged.
(1) The first is to just use an electric drill to remove the head of the screw. Drill directly through the center of the head with a drill bit about 1/3 the size of the screw's head, about 1/8" deep. Use progressively larger bits until the head of the screw comes off. Then, remove the maintenance panel. There should be more than enough screw shank (the "body" of the screw) remaining to grip with a pair of pliers... preferable self locking pliers such as Vice Grips. Hold tight and carefully turn the screw out.
(2) If course, if the panel you are removing is very thin, there might not be enough screw shank to grip… so you can use a screw extractor instead. A screw extractor looks something like a blunt-ended screw with very broad threads. These hardened cutting threads are in the opposite direction of a standard right-hand threaded screw. When you turn the extractor, it tightens into the hole giving it a firm grip. To get a correctly sized screw extractor, take one of the screws you successfully removed to the hardware store. The salesman should be able to give you the correct size extractor and the right size drill bit for this job! The hole size is important… it should be no more than about 75% of the diameter of the screw shank. If you attempt to use too large an extractor by drilling too large a hole you may overly weaken the screw shank and the head might just break off! Back to step one!
To remove the screw, drill into the damaged screw head about 1/8-1/4 inch deep. Push the extractor into the hole and rotate it counterclockwise with pliers or a Vise Grips to remove the screw.
(3) There is another possible way to get the screw out that is a little less laborious but has a lower success rate, especially if the screw is really jammed into place. Instead of trying to remove the screw with a Phillips screwdriver, try using a square-drive bit instead. Square-drive bits are designed to be used with a special type of screw used in both construction and cabinetmaking. The head of the screw has a deep square moulded into it. This design gives tremendous gripping power to the bit and is more difficult to strip.
Anyway, sometimes it is possible to tap a square-drive bit into a damaged Phillips screw head and get just enough grip to extract it! If you don't have a screwdriver with replaceable tips (everyone should have one), you can get one at the hardware store… plus an assortment of square drive bits to go with it!
That's it! Choose wisely!